H.BagherpourandM.Saeedhosseinidownloads.hindawi.com/archive/2013/519842.pdf · E155 2 < Q 2 < 13.1...

Hindawi Publishing CorporationISRN High Energy PhysicsVolume 2013, Article ID 519842, 5 pageshttp://dx.doi.org/10.1155/2013/519842

Research ArticleTransverse Spin Structure Function 𝑔

2(𝑥,𝑄

2)

H. Bagherpour and M. Saeedhosseini

Physics Department, Semnan University, Semnan, Iran

Correspondence should be addressed to H. Bagherpour; [email protected]

Received 16 April 2013; Accepted 9 May 2013

Academic Editors: F. R. Joaquim, A. Koshelev, and W. Li

Copyright © 2013 H. Bagherpour and M. Saeedhosseini. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

The transverse spin-dependent nucleon structure function 𝑔2(𝑥, 𝑄2) at large𝑄2 is computedwithin theWandzura-Wilczek relation,

using NLO 𝑔1fit. Also we investigate the twist-3 contribution of 𝑔

2(𝑥, 𝑄2) via calculating 𝑑

2. It turns out that although the twist-3

part of 𝑔2(𝑥, 𝑄2) can be negligible for DIS processes, it has a significant value in resonance region.

1. Introduction

Over the past 30 years, significant progress has been madein understanding the spin structure of the nucleon throughmeasurements using polarized deep inelastic lepton scat-tering (DIS). Most of these experiments were focused onprecise measurements of the spin structure function 𝑔

1. In

the naive quark parton model (QPM), 𝑔1is directly related

to contributions of the individual quark flavors to the overallspin of the nucleon. Sum rules based on this simple modelhave provided fertile ground for understanding the origin ofthe nucleon spin in terms of quark degrees of freedom. Inaddition, next-to-leading-order (NLO) analyses of the world𝑔1data (see, e.g., [1–5]) have provided indirect information

about the role of gluons in the nucleon’s spin. PolarizedDIS also provides information about a second spin structurefunction, 𝑔

2, which is identically zero in the naive QPM [6].

Interest in 𝑔2arises because, unlike 𝑔

1, contributions from

certain nonperturbative QCD processes such as quark-gluoncorrelations enter at the same order in 𝑄2 as asymptoticallyfree contributions [7].

2. Theoretical Background

2.1. The 𝑔2Structure Function. The nucleon’s second polar-

ized structure function 𝑔2(𝑥, 𝑄2) had never been measured

till 1990, and there had been a few theoretical studies of it.If we define 𝑞

𝑓(𝑥) and 𝑞

𝑓(𝑥) as the expectation value for the

number of quarks and antiquarks of flavor 𝑓 in the hadron

whosemomentum fraction lies in the interval [𝑥; 𝑥+𝑑𝑥], thenin the parton model it can be shown that

𝐹1(𝑥) =

1

2∑

𝑓

𝑒2

𝑓[𝑞𝑓(𝑥) + 𝑞

𝑓(𝑥)] ,

𝑔1(𝑥) =

1

2∑

𝑓

𝑒2

𝑓[𝑞𝑓(𝑥) − 𝑞

𝑓(𝑥)] ,

(1)

where the quark charge 𝑒𝑓enters due to the fact that the

cross section is proportional to the squared charge of thetarget. The Callan-Gross [8] relation shows that 𝐹

2can be

defined entirely in terms of 𝐹1, but there is no such simple

physical interpretation of 𝑔2. This spin-dependent structure

function is determined by the 𝑥-dependence of the quarkstransverse momenta and the off-shellness, both of which areunknown in the partonmodel [9]. Ignoring quarkmass effectof order O(𝑚

𝑞/∧QCD), 𝑔2 can be separated into leading twist

and higher-twist components as follows:

𝑔2(𝑥, 𝑄2) = 𝑔𝑊𝑊

2(𝑥, 𝑄2) + 𝑔2(𝑥, 𝑄2) , (2)

where

𝑔2(𝑥, 𝑄2) = −∫

1

𝑥

𝜕

𝜕𝑦[

𝑚𝑞

𝑀ℎ𝑇(𝑦, 𝑄2) + 𝜁 (𝑦, 𝑄

2)]

𝑑𝑦

𝑦. (3)

To twist-3, there are three contributions to 𝑔2[10]:

𝑔𝑤𝑤

2: the leading twist-2 term, which depends only on

𝑔1,

2 ISRN High Energy Physics

ℎ𝑇: arises from the quark transverse polarization dis-

tribution. Also twist-2, this term, is suppressed by thesmallness of the quark mass,

𝜁: the twist-3 part which arises from quark-gluoninteractions,

Using the fact that 𝑔1and 𝑔

2contain the same twist-2

operator, Wandzura-Wilczek [11] derived the followingexpression for the asymptotically free contribution to 𝑔

2, in

terms of 𝑔1:

𝑔𝑤𝑤

2(𝑥, 𝑄2) = −𝑔

1(𝑥, 𝑄2) + ∫

1

𝑥

𝑔1(𝑥, 𝑄2)

𝑥𝑑𝑥. (4)

Also this relation can be extracted directly from twist-2operators of OPE.

2.2. Operator Product Expansion. An appropriate formalismto study 𝑔

2beyond the simple QPM is the operator product

expansion (OPE) [13, 14] which is the model-independentapproach based directly on QCD.The OPE allows us to writethe hadronicmatrix element in deep inelastic scattering (DIS)in terms of a series of normalized operators of increasingtwist. The leading contribution is twist-2, with higher-twistterms suppressed by powers of 1/𝑄2. Keeping only terms upto twist-3, themoments of 𝑔

1and 𝑔2at fixed𝑄2 can be related

to the twist-2 and twist-3 reduced matrix elements, 𝑎𝑛and 𝑑

𝑛

[14],

∫

1

0

𝑥𝑛𝑔1(𝑥, 𝑄2) 𝑑𝑥 =

𝑎𝑛

2, 𝑛 = 0, 2, 4, . . . ,

∫

1

0

𝑥𝑛𝑔2(𝑥, 𝑄2) 𝑑𝑥 =

1

2

𝑛

𝑛 + 1(𝑑𝑛− 𝑎𝑛) , 𝑛 = 2, 4 . . . .

(5)

In the expression above, 𝑑𝑛directly appears in the

equation for 𝑔2allowing us to study the higher-twist structure

of the nucleon at leading order. One obtains

∫

1

0

𝑥𝑛[𝑔1(𝑥, 𝑄2) +

𝑛 + 1

𝑛𝑔2(𝑥, 𝑄2)] 𝑑𝑥

=1

2𝑑𝑛(𝑄2) , 𝑛 = 2, 4, 6 . . . .

(6)

Hence, for 𝑛 = 2,

𝑑2= 2∫

1

0

𝑥2[𝑔1(𝑥, 𝑄2) +

3

2𝑔2(𝑥, 𝑄2)] 𝑑𝑥. (7)

At large 𝑄2, the 𝑑

𝑛matrix element is related to the

color polarization, which describes how the color electricand magnetic fields respond to the nucleon spin. At lowermomentum transfer, 𝑑

𝑛(𝑄2) provides a mean to study the

transition fromperturbative to nonperturbative behavior andto quantify higher-twist effects. And although 𝑑

𝑛is a higher-

twist OPE object, the definition holds for all 𝑄2.

0.01 0.1 1x

xgP 2

(x,Q

2)

0.10.080.060.040.02

0−0.02

−0.04

−0.06

−0.08

−0.1

ModelE143 1.49 < Q

2< 8.85

E155 1.1 < Q2< 8.3

Figure 1: The polarized structure function 𝑥𝑔𝑝2as a function of 𝑄2

in intervals of 𝑥.

0.01 0.1 1x

xgd 2

(x,Q

2)

0.1

0.08

0.06

0.04

0.02

0

−0.02

−0.04

−0.06

−0.08

−0.1

Model Q2= 5 GeV2

E143 1.49 < Q2< 8.86 GeV2

2E155 2 < Q

2< 13.1 GeV

Figure 2: The polarized structure function 𝑥𝑔𝑑2as a function of 𝑄2

in intervals of 𝑥.

3. Formalism

We summarize the key features of our QCD analysis here.Knowing the fact that world data for 𝑔

1cover a broad range

in 𝑥 and 𝑄2 with relatively high precision motivates us to

extract polarized parton distributions from NLO fits to thedata. In our calculation, the polarized parton densities areparameterized at a starting scale𝑄2

0= 4GeV2and are evolved

to higher factorization scales using a numerical solution ofthe polarized NLO DGLAP evolutions [15–17]. We take theQCD coupling as a free parameter which is fitted to the data.The QCD DGLAP equations are solved in the Mellin space.

The centerpiece of our approach is the Jacobi polynomialexpansion; this method was developed and applied to a

ISRN High Energy Physics 3

xg2

Q2

Proton x = 0.039

x = 0.138

x = 0.245

x = 0.38

x = 0.68 x = 0.68

x = 0.38

x = 0.245

x = 0.138

x = 0.039Deuteron

1 2 3 4 5 6 7 8 9 10 201 2 3 4 5 6 7 8 9 10 20

0.1

0.1

0.03

0

0

0

0

0

−0.04

0.04

−0.04

−0.03

−0.06

−0.1

Figure 3: 𝑥𝑔2for the proton and deuteron as a function of𝑄2 for selected value of 𝑥. Data are for E155 [12] experiment.The error bars shown

are the statistical and systematic ones added in quadrature. The curves show 𝑥𝑔𝑊𝑊

2based on model (solid) and the BB model (dash-dot) [2].

variety of different QCD analyses for unpolarized [18, 19] andpolarized applications [4, 5]. In this approach, we expand thestructure functions in terms of the Jacobi polynomials Θ𝛼,𝛽

𝑛

𝑥𝑔𝑁max1

= 𝑥𝛽(1 − 𝑥)

𝛼

𝑁max

∑

𝑛=0

Θ𝛼,𝛽

𝑛(𝑥)

𝑛

∑

𝑗=0

𝑐(𝑛)

𝑗(𝛼, 𝛽) [𝑥𝑔

1, 𝑗 + 2] .

(8)

This relates the 𝑔1(𝑥, 𝑄2) structure function to their

moments [4].In this analysis we use 𝑁max = 9, 𝛼 = 3.0, and

𝛽 = 0.5. The world data are at relatively large 𝑄2,

where higher-twist effects should be negligible; therefore theevolved parton distributions allow one to calculate the twist-2 spin-dependent structure functions 𝑔

1and 𝑔𝑊𝑊

2, in most

kinematic regions accessible today. Due to the fact that ourcalculations are performed in DIS region where transversespin is a suppressed leading twist, we can neglect the twist-3 part of 𝑔

2, 𝜁(𝑦, 𝑄2). Also, ignoring the masses of three

light quarks is another fact that allows us to have an estimateof𝑔2, by these approximations, 𝑔

2≈ 0 then 𝑔

𝑊𝑊

2(see (2),

(3)). By the last assumption, we can see that the contributionof quark transverse distribution, ℎ

𝑇, is omitted too. Precise

measurements of 𝑔2at specific values of 𝑥 and 𝑄

2 can becompared to 𝑔𝑊𝑊

2, providing a unique opportunity to cleanly

isolate higher-twist contributions.

4. Results

Wehave performed a QCD analysis of the inclusive polarizedDIS data at NLO and extracted the spin structure function𝑔1(𝑥, 𝑄2). We have used an expansion in the Jacobi polyno-

mials to facilitate the analysis.The Jacobi polynomials help usto produce 𝑔

1(𝑥, 𝑄2) in an analytical (not numerical) form.

For the measured region 0.03 < 𝑥 < 0.8, proton anddeuteron spin structure functions 𝑔𝑊𝑊

2(𝑥, 𝑄2) are calculated

using our parametrization on 𝑔1(𝑥, 𝑄2) and the possible

contributions of higher-twist effects to 𝑔𝑝,𝑑

2are quantified

4 ISRN High Energy Physics

Table 1: Comparison of model results with proton and deuterondata.

Paper 𝑑(2)

𝑝𝑑(2)

𝑑

This paper 1.3 × 10−2

− 8.8 × 10−3

c.m. bag model [21] 1.74 × 10−2

6.79 × 10−3

MIT bag model [22, 23] 1.0 × 10−2

5.0 × 10−3

QCD sum rule [24] − (0.6 ± 0.3) × 10−2

− (17 ± 5) × 10−3

QCD sum rule [25] − (0.3 ± 0.3) × 10−2

− (13 ± 5) × 10−3

Data [26] (0.32 ± 0.17 ) × 10−2

(7.9 ± 4.8) × 10−3

Lattice QCD [27] − (4.8 ± 0.5) × 10−2

− (22 ± 6) × 10−3

using twist-3 reduced matrix element 𝑑2at fixed𝑄2. By using

(7) and substitution 𝑔𝑊𝑊2

for 𝑔2, as discussed, we neglect the

contribution from the region 0 ≤ 𝑥 < 0.03 because of the𝑥2 suppression factor. For 0.8 < 𝑥 ≤ 1, we assume that

both 𝑔1and 𝑔

2behave as (1 − 𝑥)3. We obtained the value of

𝑑(2)

𝑝= 1.3 × 10

−2 and 𝑑(2)𝑑

= − 8.8 × 10−3. The computational

processes were performed at 𝑄2 = 5GeV2. The consistencyof these values with zero is another reasonable proof forour assumptions in the most common kinematic region.Comparisons of our results for the third moments of 𝑔

2with

recent data and other model predictions are listed in Table 1.Figure 3 shows 𝑥𝑔

2for the proton and deuteron as a

function of 𝑄2 for selected value of 𝑥. Data are for E155experiment. The error bars shown are the statistical andsystematic ones added in quadrature.The curves show 𝑥𝑔

𝑊𝑊

2

based on our model (solid) and the BB model (dash-dot),both using NLO 𝑔

1fit, which are in good agreement with

the data. 𝑥𝑔2from E143 (square) [20] and E155 (circle) [12]

experiments together with our twist-2 𝑔𝑊𝑊2

at 𝑄2 = 5GeV2

for 𝑥𝑔𝑝2and 𝑥𝑔𝑑

2are presented in Figures 1 and 2.

Disclosure

This paper was published in PACS, number: 13.88.+e.

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ISRN High Energy Physics 5

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